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A Fluid Limit for the Engset Model
An Application to Retrial Queues
Stylianos Georgiadis
1
, Pascal Moyal
1
, Tam´as B´erczes
2
and J´anos Sztrik
2
1
Laboratoire de Math´ematiques Appliqu´ees de Compi`egne, Universit´e de Technologie de Compi`egne,
Centre de Recherches de Royallieu, BP 20529, 60205, Compi`egne, France
2
Faculty of Informatics, University of Debrecen, Egyetem tr 1, P.O. Box 12, 4010 Debrecen, Hungary
Keywords:
Engset Model, Fluid Limit, Semi-martingale Decomposition, Retrial Queues.
Abstract:
We represent the classical Engset-loss model by the stochastic process counting the number of customers in
the system. A fluid limit for this process is established for all the possible values of the various parameters
of the system, as the number of servers tends to infinity along with the number of sources. Our results are
derived through a semi-martingale decomposition method. A numerical application is provided to illustrate
these results. Then, we represent a finite-source retrial queue considering in addition the number of sources in
orbit. Finally, we extend the fluid limit results to a retrial queueing system, discussing different cases.
1 INTRODUCTION
In many real-life queueing systems of finite capacity,
a customer may find a full system upon arrival. In sev-
eral finite-source models, this requestcan return to the
source and stay there for a randomly distributed time
until it tries again to reach a server. The Engset model
represents a loss queueing system having this input
mechanism for several finite sources producing Pois-
son processes of the same intensity (see, e.g., (Engset,
1918)). We suppose that the system has no buffer,
hence a request is either immediately served or im-
mediately lost, whenever no server is available upon
arrival.
Such a model has been applied to a variety of re-
alistic computer and telecommunication systems and
networks. For exemple, an Engset system is adequate
to represent a radio-mobilenetwork in which the radio
sources emit messages only if no message of the same
source is currently in service. One could think that the
radio sources re-emit the same message as long as the
latter is refused due to the fact that all channels are
busy, and wait to re-issue a new message whenever
the previous message is in treatment.
This model has a wide field of applications, so it
has been studied extensively through analytical and
algorithmic methods as well. However, when the sys-
tem becomes very large, several complexity problems
may appear. The fluid limit technique offers the possi-
bility to approximate the exact values of some charac-
teristics of the system, when one or more parameters
tend to infinity. In our case, the number of servers
tends to infinity along with the number of sources.
Such techniques have been applied fruitfully to many
queueing systems (Robert, 2000; Asmussen, 2003;
Anisimov, 2007; Decreusefond and Moyal, 2012).
Recently, (Feuillet and Robert, 2012) constructed ex-
ponential martingales for the Engset model, allow-
ing to derive asymptotic estimates for several hitting
times of interest. We build on these results to de-
rive the fluid limit of an Engset model having a single
server (Section 3), and then several servers (Section
4). Simulations are presented in Section 5.
In a finite source retrial queue, the messages
which could not reach a serverare sent to the so-called
orbit, from which they are re-emitted on and on, at a
rate that is possibly higher than the original one. It is
then easily seen that the Engset model in nothing but a
particular case of a retrial queueing system for which
the two emission rates are equal. Based on this obser-
vation, in Section 6 we investigate some applications
of our initial result to derive the fluid approximation
of a retrial queue, under various conditions on the sys-
tem parameters.
2 THE ENGSET MODEL
We consider an Engset system with S (S ≥ 1) servers.
There are K (K > S) independent Poisson sources
emitting requests with intensity λ. The service times
315
Georgiadis S., Moyal P., Bérczes T. and Sztrik J..
A Fluid Limit for the Engset Model - An Application to Retrial Queues.
DOI: 10.5220/0004203101170122
In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 117-122
ISBN: 978-989-8565-40-2
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)